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Facets and layers of function for college students in beginning algebra

By Phil DeMarois


The first mathematics course for approximately 53 percent of U.S. community college\ud students is a developmental algebra course. Many such students appear to be severely\ud debilitated by their previous encounters with mathematics. Due to numerous misconceptions\ud that dictate against a traditional course, a "reform" beginning algebra course,\ud with function as the unifying concept, was designed. Since there is little research on\ud this population to justify such a approach, the key research question for this thesis\ud becomes: Can adult students who arrive at college having had debilitating prior experiences\ud with algebra acquire at least a process level understanding of function through\ud appropriate instructional treatment? Answering this question provides crucial information\ud for future curricular design in the area of developmental mathematics at the college\ud level.\ud The theoretical framework considers different aspects that make up the function concept,\ud taking critical account of several current theories of multiple representations and\ud encapsulation of process as object to build a view of function in terms of different facets\ud (representations) and different layers (of development via procedure, process,\ud object, and procept).\ud Ninety-two students at four community colleges completed written function surveys\ud before and after a "reform" beginning algebra course. Twelve students, representing all\ud four sites, participated in task-based interviews. Comparison of pre- and post-course\ud surveys provided data indicating statistically significant improvement in student abilities\ud to correctly interpret and manipulate function machines, two-variable equations,\ud two-column tables, two-dimensional graphs, written definitions and function notation.\ud The students were divided into three categories (highly capable, capable, and incapable)\ud based on their demonstrated understanding of function. Using the interviews, visual\ud profiles for students in each category were developed. The profiles indicate that the\ud development of the concept image of function in such students is complex and uneven.\ud The cognitive links between facets is sometimes nonexistent, sometimes tenuous, and\ud often unidirectional. The highly capable demonstrated some understanding across all\ud facets while the incapable indicated understanding of the more primitive facets, such\ud as colloquial and numeric, only. Profound differences were noted particularly in the\ud geometric, written, verbal, and notation facets. Overall, the target population appeared\ud able to develop a process layer understanding of function, but this development was far\ud from uniform across facets and across students

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